Computational notebook for PhD Seminar

Statistical method of averaging

• Consider the simplistic but most popular measure of averaging
• Suppose we observe a set of the price change path phenomenon of a stock over t periods \(x_1, x_2,...,x_t\).

[1] 112.2274

• The averaging function highlighted above is an algorithm and is represented by the red line in the plot opposite

Fake Data and the average as an estimate

• The mean value summarises the path price into a single number
• But how accurate is this algorithm?
• The textbook answer is given in terms of the standard error

Standard error function

[1] 0.98525

• Here averaging is the .content-box-red[algorithm], while the standard error provides the .content-box-blue[inference] of the algorithm’s accuracy.

Explanation

• It is a surprising, and crucial, aspect of statistical theory that the same data that supplies an estimate can also assess its accuracy.
• Strictly speaking Inference concerns more than accuracy: recall that algorithms say what the statistician does while inference says why she does it.

Algorithms and Inference

• Of course, the standard_error() function defined previously is itself an algorithm, which could be (and is) subject to further inferential analysis concerning its accuracy

• .content-box-green[The point is that the algorithm comes first and the inference follows at a second level of statistical consideration.]

• .content-box-yellow[In practice this means that algorithmic invention is a more free-wheeling and adventurous enterprise]

• .content-box-grey[In contrast inference playing catch-up as it strives to assess the accuracy, good or bad, of some hot new algorithmic methodology.]

Algorithms and regression

.panelset[ .panel[.panel-name[Least squares algorithm for linear regression] - The least squares estimator is a popular algorithm for estimating a linear regression - The algorithm fits the data by least squares, by minimising the sum of squared deviations over all choices of the model parameters. - Consider the following fake relationship between the price and some market factor

Least squares algorithm

• This code manufactures a positive relationship to the factor plus some noise and then draws the least squares regression line.
• The accuracy of this estimate is given by \(\pm\) 2 standard errors
• The appropriate inference of this banded grey area is that this has a 95% chance in including the true expected value of Price in an Up market.
• This 95% coverage depends on the validity of the linear regression model, which could as easy have been a quadratic relationship

Lowess algorithm for localised regression

• Lowess is a modern computer based algorithm which works by moving its attention along the x-axis, fitting local polynomial curves of differing degrees to nearby (x,y) coordinates.
• The fitted estimate above has a similar linear regression as the least squares algorithm in the middle of the data but for higher values of the factor has a much steeper curve.